$$\lim_{n\rightarrow\infty}{\left(1-\frac{1}{2^n}\right)}^{2^n}=\frac{1}{e}$$ which you can prove.

Added : It is not so difficult to prove it. Just substitute $x=2^n$ and note that $x\rightarrow \infty $ as $n\rightarrow \infty$. Also note that:
$$\lim_{x\rightarrow\infty}\left(1+\frac{(-1)}{x}\right)^x=e^{-1}$$